notes
these are some more informal notes that I have made after having taken the rigorous Analysis course.
- $c_{00}$ can be thought of as 'finite vectors'. all $\mathbb{R}^n$ tuples can be written as elements of $c_{00}$ with infinitely many zeros after the $n$th term.
- $c_0$ are all sequences that converge to zero. thus they will include all those in $c_{00}$ and more.
- $\ell^2$ are the vectors that fade fast enough for their energy 𐃏 to stay finite
- $\ell^\infty$ are the infinite vectors that never blow up – their entries are bounded.
\[c_{00} \subset c_0 \subset l^\infty \]
I am finding Real Analysis to be more difficult than any other mathematics that I have studied before. I can seem to verify the truth of statements because they seem right; but I am having a difficult time producing rigorous and correct proofs.
It seems that High-School children (on the internet) are able to self-study Fomin with success. Bitterly, we remind ourselves:
"Comparison is the thief of Joy"—Theodore Roosevelt (probably)